CN110545179A - R-LWE-based NTRU encryption method and security proving method thereof - Google Patents
R-LWE-based NTRU encryption method and security proving method thereof Download PDFInfo
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Abstract
the invention discloses an NTRU encryption method based on R-LWE and a security proving method thereof, wherein the defect that the NTRU lacks strict security proving is solved by properly modifying the structure of the NTRU ring, acquiring a key pair through defining parameters and Gaussian sampling, encrypting and decrypting, and the provable NTRU encryption method based on the R-LWE problem is further verified by proving the security of the NTRU encryption method. The NTRU encryption method based on the R-LWE and the security proving method thereof can effectively prove the security of the encryption method, increase the screening function when generating the key pair and effectively avoid lattice attack.
Description
Technical Field
the invention relates to the field of encryption methods, in particular to an NTRU encryption method based on R-LWE and a security proving method thereof.
background
the idea of the public key cryptosystem was proposed in 1976: the key is divided into a public key and a secret key, and the reciprocal encryption mapping and decryption mapping are respectively determined. Under this cryptosystem, each user has its own public and secret keys. The public key is public and can be consulted like a telephone number, so that the two communication parties can carry out secret communication without stipulating in advance and the problem of transmitting the key by a 'safety channel' does not exist; the secret key is secret and is stored by each user for decryption. From the mathematical difficulty problem, the public key cryptosystem has the following general categories: (1) public key cryptosystems based on integer factorization problems, such as RSA, Rabin, LUC, Williams, etc.; (2) public key cryptosystems designed based on the problem of discrete logarithm over finite fields, such as Diffie-Hellman key exchange protocol, ELGamal encryption system, DSA digital signature system, etc.; (3) public key cryptosystems based on discrete logarithm problems on elliptic curves, such as an ECIES system, an ECDSA system and the like; (4) the public key cryptosystem based on the knapsack problem, such as Merkle-Hellman knapsack encryption system, Chor-Rivest knapsack encryption system, etc.
at present, there are two main types of mature and widely used cryptosystems: one is RSA designed based on integer decomposition problems; the other is cryptosystem ECC (elliptic Curves cryptography) designed by discrete logarithm problem on elliptic curve.
The lattice cipher, which is a typical representative of the post-quantum-era public key cipher, has the advantages of being resistant to quantum computer attack, having average security that can be established on the complexity of the worst case lattice problem, having a linear structure, and expecting a faster encryption and decryption speed, and the like, and has attracted more and more attention in recent years. However, the main disadvantages of the general lattice public key cryptography scheme are large amount of keys, large space complexity, etc., which make the lattice public key cryptography scheme less efficient to calculate and difficult to apply in practice in large quantities.
to solve the problem of low efficiency of LWE-based schemes, Lyubash et al, 2010, proposed a new LWE, i.e., a Ring-Error Learning R-LWE (Ring-Learning With Error) problem, the following definitions are given:
for a given positive integer n, the prime number q is 1mod 2n, the integer ring Rq is z (x)/(xn +1), the error probability distribution Ψ on the ring Rq, and any number of elements (ai, bi) ═ i (ai, (ai, s) + ei) in the R-LWE distribution As, Ψ are selected.
The R-LWE search problem targets are: outputting an element satisfying the equation bi ═ i (ai, s) + ei with a non-negligible probability
The R-LWE decision problem targets are: discriminating with non-negligible probability whether the elements (ai, bi) are taken from the LWE distribution above or are uniformly distributed
Currently, the NTRU encryption scheme is the only widely used lattice cipher scheme, which is now accepted by the IEEE P1363.1 standard and the X9.98 standard, but the existing NTRU encryption scheme does not have the property of being provably secure, and it may cause lattice attacks in the key generation algorithm.
disclosure of Invention
The invention aims to provide an NTRU encryption method based on R-LWE and a security proving method thereof aiming at the defects in the prior art, which can effectively prove the security of the encryption method, increase a screening function when generating a key pair and effectively avoid lattice attack.
In order to achieve the above object, in one aspect, the present invention provides an NTRU encryption method based on R-LWE, including the following steps:
firstly, defining parameters:
defining n as the power of 2 and n is more than or equal to 8;
Definition p ═ 2;
defining q as prime number, q is more than or equal to 5, q is less than or equal to poly (n), and q is 1mod 2n to ensure that ((xn +1) modq) has n different linear factors;
the operation result of mod q is limited, and the operation result of mod (xn +1) is a polynomial with the degree not exceeding n-1;
definition of ring R ═ z (x)/(xn +1), Rq ═ R/q;
secondly, generating a key pair:
obtaining a private key through gaussian sampling in a ring R ═ z (x) (xn +1), comprising the steps of:
(1) inputting parameters n, q, p, a deviation vector s in Gaussian sampling and a safety parameter SePa, wherein q belongs to Z and represents a set formed by all reversible elements in Rq, the deviation vector s belongs to Rq, and the safety parameter is used for screening a private key obtained by sampling;
(2) Sampling f 'from the Gaussian distribution, making f equal to p · f' +1, and if so, resampling; if the calculated f does not satisfy | | | f | | | is less than or equal to SePa, resampling is needed;
(3) sampling g from the Gaussian distribution, and if so, resampling; if the sampled g does not satisfy | | | g | | | is less than or equal to SePa, resampling is needed;
(4) Let the private key sk be f, the public key pk be h be pg/f, and the public key outputs the key pair
thirdly, encryption:
adding a random deviation vector s belonging to Rq in the encryption process, wherein e belongs to psi alpha, and outputting a ciphertext c belonging to h · s + pe + m mod q;
Fourthly, decryption:
decrypting the normally received ciphertext c by using the private key sk and outputting a plaintext m, and the method specifically comprises the following steps:
f·c＝f·h·s+f·pe+f·m mod q＝pg·s+f·pe+f·m mod q
When the coefficient of pg · s + f · pe + f · m mod q is within the interval, then further:
pg·s+f·pe+f·m mod q＝pg·s+f·pe+f·m
mod p operation is performed on pg · s + f · pe + f · m, that is:
pg·s+f·pe+f·m mod p＝f·m mod p＝m
Further, the distribution of the keys pk — h in a uniform distribution is statistically indistinguishable.
On the other hand, the invention also provides a security certification method of the NTRU encryption method based on the R-LWE, which comprises the following steps:
a: given an attack algorithm A for selecting plaintext attack, then constructing an algorithm B for solving an R-LWE problem:
b: providing an interrogator O, with which the algorithm B extracts a sample (h ', c') from the uniform or LWE distribution, where c '═ h's + e;
c: computing a public key h ═ ph '∈ Rq of the algorithm A by using a public key h' of the algorithm B;
d: when algorithm a presents challenge plaintext information m0, m1, algorithm B randomly picks out B from the uniform distribution U (0, 1);
e: calculating the ciphertext c of the algorithm A as pc ' + mb epsilon to Rq, returning the ciphertext c to the algorithm A, outputting a guess B ' of B by the algorithm A, and outputting 1 by the algorithm B if B is B ', so that the NTRU encryption method is proved to be safe; if B ≠ B', algorithm B outputs 0, proving that the NTRU encryption method is insecure.
further, since it is known that the public key h' used by the algorithm B and the public key h used by the algorithm a are uniformly random in the middle, and the statistical distance between the public key h in the algorithm a and the public key of the real attack is within q- Ω (n).
further, since c 'is h's + e and s e Rq, e Ψ α, the distribution of the ciphertext c returned to algorithm a is the same as under the chosen plaintext attack.
further, if the interrogator O outputs samples in the LWE distribution, algorithm A attacks successfully and algorithm B takes the output as 1.
further, if the interrogator O outputs samples in a uniform distribution, since then the values of pc' and c are uniformly random in Rq and are independent of B, the probability of algorithm B is 1/2, and the output is 1.
Compared with the prior art, the invention has the advantages that: a provable NTRU encryption method based on R-LWE problem solves the defect that NTRU lacks strict security certification by properly modifying the structure of NTRU ring, improves security, adds a 'screening' function to the key generated by direct sampling in the process of key generation algorithm, effectively avoids lattice attack, and makes the scheme capable of being applied to practice.
Detailed Description
The following non-limiting detailed description of the present invention is provided in connection with the preferred embodiments.
An R-LWE based NTRU encryption method, comprising the following steps:
firstly, defining parameters:
Defining n as the power of 2 and n is more than or equal to 8;
definition p ═ 2;
defining q as prime number, q is more than or equal to 5, q is less than or equal to poly (n), and q is 1mod 2n to ensure that ((xn +1) modq) has n different linear factors;
The operation result of mod q is limited, and the operation result of mod (xn +1) is a polynomial with the degree not exceeding n-1;
Definition of Ring to elements in Ring R
f (x) -f 0+ f1x + … + fn-1xn-1, g (x) -g 0+ g1x + … + gn-1xn-1 and k e Z, defining the operation:
k·f(x)＝kf+kfx+…+kfx
secondly, generating a key pair:
obtaining a private key through gaussian sampling in a ring R ═ z (x) (xn +1), comprising the steps of:
(1) Inputting parameters n, q, D, a deviation vector s in Gaussian sampling and a security parameter SePa, wherein q belongs to Z and represents a set formed by all reversible elements in Rq, the deviation vector s belongs to Rq, the security parameter is used for screening private keys obtained by sampling, in order to efficiently obtain private key pairs meeting discrete Gaussian distribution, the selected deviation parameter is usually very large, so that the size of the private key pairs is very large sometimes, effective lattice attack is caused, and the security parameter SePa is set to improve the security;
(2) sampling f 'from the Gaussian distribution, making f equal to p · f' +1, and if so, resampling; if the calculated f does not satisfy | | | f | | | is less than or equal to SePa, resampling is needed;
(3) sampling g from the Gaussian distribution, and if so, resampling; if the sampled g does not satisfy | | | g | | | is less than or equal to SePa, resampling is needed;
(4) let the private key sk be f, the public key pk be h be pg/f, and the public key outputs the key pair
therefore, the method not only ensures the cryptographic property of the pseudo-randomness of the public key, but also has the function of screening the sampled and selected key.
thirdly, encryption:
adding a random deviation vector s belonging to Rq in the encryption process, wherein e belongs to psi alpha, and outputting a ciphertext c belonging to h · s + pe + m mod q;
fourthly, decryption:
Decrypting the normally received ciphertext c by using the private key sk and outputting a plaintext m, and the method specifically comprises the following steps:
f·c＝f·h·s+f·pe+f·m mod q＝pg·s+f·pe+f·m mod q
When the coefficient of pg · s + f · pe + f · m mod q is within the interval, then further:
pg·s+f·pe+f·m mod q＝pg·s+f·pe+f·m
Mod p operation is performed on pg · s + f · pe + f · m, that is:
pg·s+f·pe+f·m mod p＝f·m mod p＝m
the following theorem is required for the NTRU encryption method to prove security:
Theorem 1: defining n as a power of 2 and n ≧ 8, q ≧ 5, q ≦ poly (n) and q ═ 1mod 2n, ε > 0, and the offset vector incorporates yi ∈ Rq and zi ═ yip-1mod q, where i ∈ {1, 2}, then the statistical distance
theorem 2: if there is a chosen plaintext attack (IND-CPA) algorithm for the encryption method with a runtime T and a success probability 1/2+ δ, then there is an algorithm with a runtime T + o (n) and a success probability δ -q- Ω (n) to solve the R-LWE problem.
From theorem 1, it can be derived that the distribution of the public key pk ═ h in the generation of the key pair in the uniform distribution is not statistically distinguishable, which provides a necessary condition for the R-LWE-based problem.
The security certification method of the NTRU encryption method based on the R-LWE comprises the following steps:
a: giving an algorithm A for selecting plaintext attack, and then constructing an algorithm B for solving an R-LWE problem;
b: providing an interrogator O, with which the algorithm B extracts a sample (h ', c') from the uniform distribution or medium, wherein c '═ h's + e;
c: computing the public key h ═ ph '∈ Ra of the algorithm A by using the public key h' of the algorithm B;
D: when algorithm a presents challenge plaintext information m0, m1, algorithm B randomly picks out B from the uniform distribution U (0, 1);
e: calculating the ciphertext c of the algorithm A as pc ' + mb epsilon to Rq, returning the ciphertext c to the algorithm A, outputting a guess B ' of B by the algorithm A, and outputting 1 by the algorithm B if B is B ', so that the NTRU encryption method is proved to be safe; if B ≠ B', algorithm B outputs 0, proving that the NTRU encryption method is insecure.
according to theorem 1, since the public key h' used by the algorithm B and the public key h used by the algorithm a are uniformly random in the middle, and the statistical distance between the public key h in the algorithm a and the public key of the real attack is within q- Ω (n). Since c 'h's + e and s e Rq, e Ψ α, the distribution of the ciphertext c returned to algorithm a is the same as under the chosen-plaintext attack. Thus, if the interrogator O outputs samples in the LWE distribution, algorithm a attacks successfully and algorithm B takes the output as 1; if the interrogator O outputs samples in a uniform distribution, then the probability of algorithm B is 1/2 and the output is 1, since then the values of pc' and c are uniformly random in Rq and are independent of B.
In summary, the scheme is secure with chosen plaintext attack (IND-CPA).
according to the NTRU encryption method based on the R-LWE and the security proving method thereof, the defect that the NTRU lacks strict security proving is overcome through appropriate modification of the structure of the NTRU ring, the security is improved, the screening function is added to the key generated by direct sampling in the process of the key generating algorithm, lattice attack is effectively avoided, and the scheme can be applied to practice.
It should be noted that the above-mentioned preferred embodiments are merely illustrative of the technical concepts and features of the present invention, and are intended to enable those skilled in the art to understand the contents of the present invention and implement the present invention, and not to limit the scope of the present invention. All equivalent changes and modifications made according to the spirit of the present invention should be covered within the protection scope of the present invention.
Claims (7)
1. an NTRU encryption method based on R-LWE is characterized by comprising the following steps:
Firstly, defining parameters:
defining n as the power of 2 and n is more than or equal to 8;
Definition p ═ 2;
Defining q as prime number, q is more than or equal to 5, q is less than or equal to poly (n), and q is 1mod 2n to ensure that ((xn +1) modq) has n different linear factors;
the operation result of mod q is limited, and the operation result of mod (xn +1) is a polynomial with the degree not exceeding n-1;
Definition of ring R ═ z (x)/(xn +1), Rq ═ R/q;
secondly, generating a key pair:
Obtaining a private key through gaussian sampling in a ring R ═ z (x) (xn +1), comprising the steps of:
(1) Inputting parameters n, q, p, a deviation vector s in Gaussian sampling and a safety parameter SePa, wherein q belongs to Z and represents a set formed by all reversible elements in Rq, the deviation vector s belongs to Rq, and the safety parameter is used for screening a private key obtained by sampling;
(2) Sampling f 'from the Gaussian distribution, making f equal to p · f' +1, and if so, resampling; if the calculated f does not satisfy | | | f | | | is less than or equal to SePa, resampling is needed;
(3) sampling g from the Gaussian distribution, and if so, resampling; if the sampled g does not satisfy | | | g | | | is less than or equal to SePa, resampling is needed;
(4) Let the private key sk be f, the public key pk be h be pg/f, and the public key outputs the key pair
Thirdly, encryption:
adding a random deviation vector s belonging to Rq in the encryption process, wherein e belongs to psi alpha, and outputting a ciphertext c belonging to h · s + pe + m mod q;
fourthly, decryption:
Decrypting the normally received ciphertext c by using the private key sk and outputting a plaintext m, and the method specifically comprises the following steps:
f·c＝f·h·s+f·pe+f·m mod q＝pg·s+f·pe+f·m mod q
when the coefficient of pg · s + f · pe + f · m mod q is within the interval, then further:
pg·s+f·pe+f·m mod q＝pg·s+f·pe+f·m
mod p operation is performed on pg · s + f · pe + f · m, that is:
pg·s+f·pe+f·m mod p＝f·m mod p＝m。
2. The R-LWE based NTRU encryption method according to claim 1, wherein: the distribution of the public key pk h in a uniform distribution is statistically indistinguishable.
3. A security certification method of the R-LWE based NTRU encryption method according to claim 1, comprising the steps of:
a: giving an attack algorithm A for selecting plaintext attack, and then constructing an algorithm B for solving an R-LWE problem;
B: providing an interrogator 0, and extracting a sample (h ', c') from the uniform distribution or the LWE distribution by the algorithm B by using the interrogator 0, wherein c 'is h's + e;
C: computing a public key h ═ ph '∈ Rq of the algorithm A by using a public key h' of the algorithm B;
d: when algorithm a presents challenge plaintext information m0, m1, algorithm B randomly picks out B from the uniform distribution U (0, 1);
e: calculating the ciphertext c of the algorithm A as pc ' + mb epsilon to Rq, returning the ciphertext c to the algorithm A, outputting a guess B ' of B by the algorithm A, and outputting 1 by the algorithm B if B is B ', so that the NTRU encryption method is proved to be safe; if B ≠ B', algorithm B outputs 0, proving that the NTRU encryption method is insecure.
4. the security certification method of the R-LWE-based NTRU encryption method according to claim 3, wherein: since the public key h' used by the algorithm B and the public key h used by the algorithm a are known to be uniformly random in the middle, and the statistical distance between the public key h in the algorithm a and the public key of the real attack is within q- Ω (n).
5. the security certification method of the R-LWE-based NTRU encryption method according to claim 4, wherein: since c 'h's + e and s e Rq, e Ψ α, the distribution of the ciphertext c returned to algorithm a is the same as under the chosen-plaintext attack.
6. the R-LWE-based NTRU encryption method for security certification according to claim 5, wherein: if interrogator 0 outputs samples in the LWE distribution, the output is 1.
7. the R-LWE-based NTRU encryption method for security certification according to claim 5, wherein: if interrogator 0 outputs samples in a uniform distribution, then the probability of algorithm B is 1/2 and the output is 1, since then the values of pc' and c are uniformly random in Rq and are independent of B.
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Application publication date: 20191206 |